Let Γ be a co-compact Fuchsian group of isometries on the Poincaré disk D and ∆ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction f of ∆, equivariant by Γ with real eigenvalue λ = −s(1 − s), where s = 1 2 + it, admits an integral representation by a distribution D f,s (the Helgason distribution) which is equivariant by Γ and supported at infinity ∂D = S 1. The geodesic flo...

The goal of this paper is to estimate the size of the first eigenfunction u uniformly for all convex domains. In particular, we will locate the place where u achieves its maximum to within a distance comparable to the inradius, uniformly for arbitrarily large diameter. In addition, we will estimate the location of other level sets of u by showing that u is well-approximated by the first eigenfu...

We investigate the mKdV hierarchy with integral type of source (mKdVHWS), which consist of the reduced AKNS eigenvalue problem with r = q and the mKdV hierarchy with extra term of the integration of square eigenfunction. First we propose a method to find the explicit evolution equation for eigenfunction of the auxiliary linear problems of the mKdVHWS. Then we determine the evolution equations o...

We study the covariant derivatives of an eigenfunction for Laplace–Beltrami operator on a complete, connected Riemannian manifold with nonzero constant sectional curvature. show that along every parallel tensor, derivative is scalar multiple eigenfunction. also polynomial depending eigenvalue and prove some properties. A conjecture motivated by vertex algebraic structure space forms announced, ...

We extend the Adomian's decomposition method to work for the general eigenvalue problems, in addition to the existing applications of the method to boundary and initial value problems with nonlinearity. We develop the Hamiltonian inverse iteration method which will provide the ground state eigenvalue and the explicit form eigenfunction within a few iterations. The method for finding the excited...

This problem is known as the Lambda Modes problem. The fundamental eigenvalue (the largest one) is called the effective multiplication, k-effective, of the reactor core. This eigenvalue and its corresponding eigenfunction describe the steady state neutron distribution in the core. On the other hand, to compute the dominant modes of the reactor core is useful to develop modal methods [2] to inte...

In this article, we study, as the coefficient s → ∞, the asymptotic behavior of the principal eigenvalue of the eigenvalue problem −φ′′(x)− 2sm′(x)φ′(x) + c(x)φ(x) = λ(s)φ(x), 0 < x < 1, complemented by a general boundary condition. This problem is relevant to nonlinear propagation phenomena in reaction-diffusion equations. The main point is that the advection (or drift) term m allows natural d...

It is shown that any μ ∈ C is an infinite multiplicity eigenvalue of the Steklov smoothing operator Sh acting on the space Lloc(R). For μ 6= 0 the eigenvalue-eigenfunction problem leads to studying a differential-difference equation of mixed type. An existence and uniqueness theorem is proved for this equation. Further a transformation group is defined on a countably normed space of initial fun...

We prove an analogue of the magnetic nodal theorem on quantum graphs: the number of zeros of the nth eigenfunction of the Schrödinger operator on a quantum graph is related to the stability of the nth eigenvalue of the perturbation of the operator by magnetic potential. More precisely, we consider the nth eigenvalue as a function of the magnetic perturbation and show that its Morse index at zer...

The Dafermos regularization of a system of n hyperbolic conservation laws in one space dimension has, near a Riemann solution consisting of n Lax shock waves, a self-similar solution u=u∈(X/T ). In Lin and Schecter (2003, SIAM J. Math. Anal. 35, 884–921) it is shown that the linearized Dafermos operator at such a solution may have two kinds of eigenvalues: fast eigenvalues of order 1/∈ and slow...